properties of the form ($l \le \pm x \pm y \leq h$) have been
found useful and effective in static analysis of numerical
programs. Their analysis serves as a key component of the tool
ASTREE based on abstract interpretation framework proposed by
Patrick Cousot and his group, which has been successfully used to
analyze commercial software consisting of hundreds of thousand of
lines of code. This talk will discuss a heuristic based on the
quantifier-elimination (QE) approach presented by Kapur (2005)
that can be used to automatically derive loop invariants
expressed as a conjunction of octagonal constraints in $O(n^2)$,
where $n$ is the number of program variables over which these
constraints are expressed. This is in contrast to the algorithms
developed in Mine's thesis which have the complexity at least
$O(n^3)$. The restricted QE heuristic usually generates
invariants stronger than those obtained by the freely available
Interproc tool. Extensions of the proposed approach to generating disjunctive
invariants will be presented.